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Theorem isoval 13667
Description: The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b  |-  B  =  ( Base `  C
)
invfval.n  |-  N  =  (Inv `  C )
invfval.c  |-  ( ph  ->  C  e.  Cat )
invfval.x  |-  ( ph  ->  X  e.  B )
invfval.y  |-  ( ph  ->  Y  e.  B )
isoval.n  |-  I  =  (  Iso  `  C
)
Assertion
Ref Expression
isoval  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )

Proof of Theorem isoval
Dummy variables  x  c  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isoval.n . . . 4  |-  I  =  (  Iso  `  C
)
2 invfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5525 . . . . . . . 8  |-  ( c  =  C  ->  (Inv `  c )  =  (Inv
`  C ) )
4 invfval.n . . . . . . . 8  |-  N  =  (Inv `  C )
53, 4syl6eqr 2333 . . . . . . 7  |-  ( c  =  C  ->  (Inv `  c )  =  N )
65coeq2d 4846 . . . . . 6  |-  ( c  =  C  ->  (
( z  e.  _V  |->  dom  z )  o.  (Inv `  c ) )  =  ( ( z  e. 
_V  |->  dom  z )  o.  N ) )
7 df-iso 13652 . . . . . 6  |-  Iso  =  ( c  e.  Cat  |->  ( ( z  e. 
_V  |->  dom  z )  o.  (Inv `  c )
) )
8 funmpt 5290 . . . . . . 7  |-  Fun  (
z  e.  _V  |->  dom  z )
9 fvex 5539 . . . . . . . 8  |-  (Inv `  C )  e.  _V
104, 9eqeltri 2353 . . . . . . 7  |-  N  e. 
_V
11 cofunexg 5739 . . . . . . 7  |-  ( ( Fun  ( z  e. 
_V  |->  dom  z )  /\  N  e.  _V )  ->  ( ( z  e.  _V  |->  dom  z
)  o.  N )  e.  _V )
128, 10, 11mp2an 653 . . . . . 6  |-  ( ( z  e.  _V  |->  dom  z )  o.  N
)  e.  _V
136, 7, 12fvmpt 5602 . . . . 5  |-  ( C  e.  Cat  ->  (  Iso  `  C )  =  ( ( z  e. 
_V  |->  dom  z )  o.  N ) )
142, 13syl 15 . . . 4  |-  ( ph  ->  (  Iso  `  C
)  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
151, 14syl5eq 2327 . . 3  |-  ( ph  ->  I  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
1615oveqd 5875 . 2  |-  ( ph  ->  ( X I Y )  =  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y ) )
17 eqid 2283 . . . . . 6  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) )  =  ( x  e.  B , 
y  e.  B  |->  ( ( x (Sect `  C ) y )  i^i  `' ( y (Sect `  C )
x ) ) )
18 ovex 5883 . . . . . . 7  |-  ( x (Sect `  C )
y )  e.  _V
1918inex1 4155 . . . . . 6  |-  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) )  e.  _V
2017, 19fnmpt2i 6193 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) )  Fn  ( B  X.  B )
21 invfval.b . . . . . . 7  |-  B  =  ( Base `  C
)
22 invfval.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
23 invfval.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
24 eqid 2283 . . . . . . 7  |-  (Sect `  C )  =  (Sect `  C )
2521, 4, 2, 22, 23, 24invffval 13660 . . . . . 6  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) ) )
2625fneq1d 5335 . . . . 5  |-  ( ph  ->  ( N  Fn  ( B  X.  B )  <->  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C )
y )  i^i  `' ( y (Sect `  C ) x ) ) )  Fn  ( B  X.  B ) ) )
2720, 26mpbiri 224 . . . 4  |-  ( ph  ->  N  Fn  ( B  X.  B ) )
28 opelxpi 4721 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
2922, 23, 28syl2anc 642 . . . 4  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
30 fvco2 5594 . . . 4  |-  ( ( N  Fn  ( B  X.  B )  /\  <. X ,  Y >.  e.  ( B  X.  B
) )  ->  (
( ( z  e. 
_V  |->  dom  z )  o.  N ) `  <. X ,  Y >. )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) ) )
3127, 29, 30syl2anc 642 . . 3  |-  ( ph  ->  ( ( ( z  e.  _V  |->  dom  z
)  o.  N ) `
 <. X ,  Y >. )  =  ( ( z  e.  _V  |->  dom  z ) `  ( N `  <. X ,  Y >. ) ) )
32 df-ov 5861 . . 3  |-  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y )  =  ( ( ( z  e.  _V  |->  dom  z )  o.  N
) `  <. X ,  Y >. )
33 ovex 5883 . . . . 5  |-  ( X N Y )  e. 
_V
34 dmeq 4879 . . . . . 6  |-  ( z  =  ( X N Y )  ->  dom  z  =  dom  ( X N Y ) )
35 eqid 2283 . . . . . 6  |-  ( z  e.  _V  |->  dom  z
)  =  ( z  e.  _V  |->  dom  z
)
3633dmex 4941 . . . . . 6  |-  dom  ( X N Y )  e. 
_V
3734, 35, 36fvmpt 5602 . . . . 5  |-  ( ( X N Y )  e.  _V  ->  (
( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y ) )
3833, 37ax-mp 8 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y )
39 df-ov 5861 . . . . 5  |-  ( X N Y )  =  ( N `  <. X ,  Y >. )
4039fveq2i 5528 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) )
4138, 40eqtr3i 2305 . . 3  |-  dom  ( X N Y )  =  ( ( z  e. 
_V  |->  dom  z ) `  ( N `  <. X ,  Y >. )
)
4231, 32, 413eqtr4g 2340 . 2  |-  ( ph  ->  ( X ( ( z  e.  _V  |->  dom  z )  o.  N
) Y )  =  dom  ( X N Y ) )
4316, 42eqtrd 2315 1  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   <.cop 3643    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   Catccat 13566  Sectcsect 13647  Invcinv 13648    Iso ciso 13649
This theorem is referenced by:  inviso1  13668  invf  13670  invco  13673  isohom  13674  oppciso  13679  funciso  13748  ffthiso  13803  fuciso  13849  setciso  13923  catciso  13939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-inv 13651  df-iso 13652
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